Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CONVERGENCE OF DOUBLE SERIES OF RANDOM ELEMENTS IN BANACH SPACES

  • Tien, Nguyen Duy (Faculty of Mathematics National University of Hanoi) ;
  • Dung, Le Van (Faculty of Mathematics Danang University of Education)
  • Received : 2011.05.20
  • Published : 2012.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

For a double array of random elements $\{X_{mn};m{\geq}1,n{\geq}1\}$ in a $p$-uniformly smooth Banach space, $\{b_{mn};m{\geq}1,n{\geq}1\}$ is an array of positive numbers, convergence of double random series ${\sum}^{\infty}_{m=1}{\sum}^{\infty}_{n=1}X_{mn}$, ${\sum}^{\infty}_{m=1}{\sum}^{\infty}_{n=1}b^{-1}_{mn}X_{mn}$ and strong law of large numbers $$b^{-1}_{mn}\sum^m_{i=1}\sum^n_{j=1}X_{ij}{\rightarrow}0$$ as $$m{\wedge}n{\rightarrow}{\infty}$$ are established.

Keywords

References

  1. L. V. Dung, Th. Ngamkham, N. D. Tien and A. I. Volodin, Marcinkiewicz-Zygmund type law of large numbers for double arrays of random elements in Banach spaces, Lobachevskii J. Math. 30 (2009), no. 4, 337-346. https://doi.org/10.1134/S1995080209040118
  2. L. V. Dung and N. D. Tien, Mean convergence theorems and weak laws of large numbers for double arrays of random elements in Banach spaces, Bull. Korean Math. Soc. 47 (2010), no. 3, 467{482.
  3. S. Gan, On almost sure convergence of weighted sums of random element sequences, Acta Math. Sci. Ser. B Engl. Ed. 30 (2010), no. 4, 1021-1028.
  4. J. I. Hong and J. Tsay, A strong law of large numbers for random elements in Banach spaces, Southeast Asian Bull. Math. 34 (2010), no. 2, 257-264.
  5. D. Landers and L. Rogge, Laws of large numbers for pairwise independent uniformly integrable random variables, Math. Nachr. 130 (1987), 189-192. https://doi.org/10.1002/mana.19871300117
  6. M. Ordonez Cabrera, Convergence of weighted sums of random variables and uniform integrability concerning the weights, Collect. Math. 45 (1994), no. 2, 121-132.
  7. G. Pisier, Probabilistic methods in the geometry of Banach spaces, Probability and analysis (Varenna, 1985), 167241, Lecture Notes in Math., 1206, Springer, Berlin, 1986.
  8. N. V. Quang, L. V. Thanh, and N. D. Tien, Almost sure convergence for double arrays of block-wise M-dependent random elements in Banach spaces, Georgian Mathematical Journal 18 (2011), 777-800. https://doi.org/10.1515/GMJ.2011.0045
  9. A. Rosalsky and L. V. Thanh, Strong and weak laws of large numbers for double sums of independent random elements in Rademacher type p Banach spaces, Stoch. Anal. Appl. 24 (2006), no. 6, 1097-1117. https://doi.org/10.1080/07362990600958770
  10. F. S. Scalora, Abstract martingale convergence theorems, Pacific J. Math. 11 (1961), 347-374. https://doi.org/10.2140/pjm.1961.11.347
  11. U. Stadtmulle and L. V. Thanh, On the strong limit theorems for double arrays of blockwise M-dependent random variables, Acta Math. Sin. (Engl. Ser.) 27 (2011), no. 10, 19231934.
  12. C. Su and T. J. Tong, Almost sure convergence of the general Jamison weighted sum of B-valued random variables, Acta Math. Sin. (Engl. Ser.) 20 (2004), no. 1, 181-192. https://doi.org/10.1007/s10114-003-0286-y
  13. N. D. Tien, On Kolmogorov's three series theorem and mean square convergence of martingales in a Banach space, Theory Probab. Appl. 24 (1980), no. 4, 797-808. https://doi.org/10.1137/1124091
  14. W. A. Woyczynski, Geometry and martingales in Banach spaces, ProbabilityWinter School (Proc. Fourth Winter School, Karpacz, 1975), pp. 229-275. Lecture Notes in Math., Vol. 472, Springer, Berlin, 1975.